A connected graph can be decomposed into its biconnected components. This block cutpoint tree is unique. Similarly, biconnected graphs can be decomposed into triconnected components. The corresponding SPQR tree describes all the 2-vertex cuts in the graph and is uniquely determined from its graph.

This process does not generalize to higher connectivity. For example, given a triconnected graph $G$, there can be multiple "trees" describing all the 3-vertex cuts of $G$.

Are there special classes of graphs such that $k$-connected graphs (in these classes) can be decomposed uniquely into their $k+1$-connected components.